The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering A.P. Stakhov International Club of the Golden Section, 6 McCreary Trail, Bolton, ON, Canada L7E 2C8 Accepted 14 January 2005 Abstract The ‘‘Dichotomy Principle’’ and the classical ‘‘Golden Section Principle’’ are two of the most important principles of Nature, Science and also Art. The Generalized Principle of the Golden Section that follows from studying the diagonal sums of the Pascal triangle is a sweeping generalization of these important principles. This underlies the foundation of ‘‘Harmony Mathematics’’, a new proposed mathematical direction. Harmony Mathematics includes a number of new mathematical theories: an algorithmic measurement theory, a new number theory, a new theory of hyperbolic functions based on Fibonacci and Lucas numbers, and a theory of the Fibonacci and ‘‘Golden’’ matrices. These mathematical theories are the source of many new ideas in mathematics, philosophy, botanic and biology, electrical and computer science and engineering, communication systems, mathematical education as well as theoretical physics and physics of high energy particles. Ó 2005 Elsevier Ltd. All rights reserved. Algebra and Geometry have one and the same fate. The rather slow successes followed after the fast ones at the beginning. They left science at such step where it was still far from perfect. It happened, probably, because Mathematicians paid attention to the higher parts of the Analysis. They neglected the beginnings and did not wish to work on such field, which they finished with one time and left it behind. Nikolay Lobachevsky 1. Introduction: ‘‘Fibonacci World’’ and ‘‘Harmony Mathematics’’ To realize the real world as a whole, to see the reflection of one and the same essence, to see one reason in the set of all life forms, to see behind the natural laws one general law, from which all of them result, is the main tendency of human culture. The idea of Harmony that descended to us from ancient science is the ‘‘launching pad’’ for the 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.038 E-mail address: [email protected]URL: www.goldenmuseum.com Chaos, Solitons and Fractals 26 (2005) 263–289 www.elsevier.com/locate/chaos
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Chaos, Solitons and Fractals 26 (2005) 263–289
www.elsevier.com/locate/chaos
The Generalized Principle of the Golden Section andits applications in mathematics, science, and engineering
A.P. Stakhov
International Club of the Golden Section, 6 McCreary Trail, Bolton, ON, Canada L7E 2C8
Accepted 14 January 2005
Abstract
The ‘‘Dichotomy Principle’’ and the classical ‘‘Golden Section Principle’’ are two of the most important principles of
Nature, Science and also Art. The Generalized Principle of the Golden Section that follows from studying the diagonal
sums of the Pascal triangle is a sweeping generalization of these important principles. This underlies the foundation of
‘‘Harmony Mathematics’’, a new proposed mathematical direction. Harmony Mathematics includes a number of new
mathematical theories: an algorithmic measurement theory, a new number theory, a new theory of hyperbolic functions
based on Fibonacci and Lucas numbers, and a theory of the Fibonacci and ‘‘Golden’’ matrices. These mathematical
theories are the source of many new ideas in mathematics, philosophy, botanic and biology, electrical and computer
science and engineering, communication systems, mathematical education as well as theoretical physics and physics
of high energy particles.
� 2005 Elsevier Ltd. All rights reserved.
Algebra and Geometry have one and the same fate. The rather slow successes followed after the fast ones at
the beginning. They left science at such step where it was still far from perfect. It happened, probably, becauseMathematicians paid attention to the higher parts of the Analysis. They neglected the beginnings and did notwish to work on such field, which they finished with one time and left it behind.
Nikolay Lobachevsky
1. Introduction: ‘‘Fibonacci World’’ and ‘‘Harmony Mathematics’’
To realize the real world as a whole, to see the reflection of one and the same essence, to see one reason in the set of
all life forms, to see behind the natural laws one general law, from which all of them result, is the main tendency
of human culture. The idea of Harmony that descended to us from ancient science is the ‘‘launching pad’’ for the
0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.
Notice that the ‘‘Dichotomy Principle’’ (3) and the ‘‘Golden Section Principle’’ (6) have a huge amount of applica-
tions in nature, science and mathematics (binary number system, numerical methods of the algebraic equation solu-
tions, sell division and so on).
In particular, the Dichotomy Principle (3) underlies the binary number system:
A ¼Xi
ai2i ð7Þ
where ai is the binary numeral 0 or 1; 2i is the weight of the ith digit of the number system (7); i = 0, ±1, ±2, ±3, . . . Thelatter underlies modern computers and information technology.
Also the Golden Section Principle (6) is the basis of Bergman�s number system [4]:
A ¼Xi
aisi ð8Þ
where ai is the binary numeral 0 or 1; si is the weight of the ith digit of the number system (8); s ¼ 1þffiffi5
p
2is the Golden
Ratio; i = 0, ±1, ±2, ±3, . . .
3. The Generalized Principle of the Golden Section
3.1. A generalization of the Golden Section
The problem of the line division in extreme and middle ratio (the Golden Section) that came to us from the ‘‘Euclidean
Elements’’ allows the following generalization. Let us give the integer non-negative number p (p = 0,1,2,3, . . .) anddivide the line AB by the point C in the following ratio (Fig. 1):
CBAC
¼ ABCB
� �p
: ð9Þ
Let us denote by x the ratio AB:CB = x; then according to (9) the ratio CB:AC = xp. On the other hand,
AB = AC + CB, from where the following algebraic equation follows:
xpþ1 ¼ xp þ 1: ð10Þ
Let us denote by sp the positive root of the algebraic equation (10).
Eq. (10) describes an infinite number of the line segment AB divisions in the ratio (9) because every p ‘‘generates’’ its
own variant of the division (9). For the case p = 0 we have: sp = 2 and then the division (9) is reduced to the classical
dichotomy (Fig. 1a). For the case p = 1 we have: sp ¼ s ¼ 1þffiffi5
p
2(the Golden Ratio) and the division (9) coincides with the
classical Golden Section (Fig. 1b). This fact is a cause why the division of the line segment in the ratio (9) was called the
generalized Golden Section or the Golden p-Section [9] and the positive roots sp of the algebraic equation (10) were called
the generalized Golden Proportions or the Golden p-Proportions [9].
(a) p = 0 A B 0=2
(b) p = 1 A B 1=1,618
(c) p = 2 A B 2=1,465
(d) p = 3 A B 3=1,380
(e) p = 4 A B 4=1,324
C
C
C
C
C
τ
τ
τ
τ
τ
Fig. 1. The Golden p-Sections (p = 0,1,2,3, . . .).
Notice that there is a fundamental distinction between the division of the line segment in Fig. 1a and the rest divi-
sions in Fig. 1b–e from the point of view ‘‘symmetry’’ and ‘‘asymmetry’’. The division in Fig. 1a is based on the
‘‘Dichotomy Principle’’ and reflects the ‘‘Symmetry Principle’’. The divisions in Fig. 1b–e are ‘‘asymmetric’’ divisions
and reflect the ‘‘Asymmetry Principle’’.
It follows from Eq. (10) the following fundamental identity that connects the adjacent powers of the Golden p-Pro-
portion sp:
snp ¼ sn�1p þ sn�p�1
p ¼ sp � sn�1p : ð11Þ
where n = 0, ±1, ±2, ±3, . . .Notice that for the cases p = 0 and p = 1 the general identity (11) is reduced to the identities (1), (4) respectively.
3.2. The Generalized Principle of the Golden Section
And now let us divide all terms of the identity (11) by snp. The following identity follows as a result of this division:
1 ¼ s0p ¼ s�1p þ s�p�1
p : ð12Þ
Using (11), (12) it is possible to construct the following ‘‘dynamic’’ model of the ‘‘Unit’’ decomposition according to
the Golden p-Proportion:
1 ¼ s0p ¼ s�1p þ s�ðpþ1Þ
p
s�ðpþ1Þp ¼ s�ðpþ1Þ�1
p þ s�2ðpþ1Þp
s�2ðpþ1Þp ¼ s�2ðpþ1Þ�1
p þ s�3ðpþ1Þp
1 ¼ s0 ¼ s�1p þ s�ðpþ1Þ�1
p þ s�2ðpþ1Þ�1p þ s�3ðpþ1Þ�1
p þ . . . ¼P1
i¼1s�ði�1Þðpþ1Þ�1p
ð13Þ
The main result of the above consideration is finding more general principle of the ‘‘Unit’’ division given by the fol-
lowing identity:
1 ¼ s�1p þ s�ðpþ1Þ
p ¼X1i¼1
s�ði�1Þðpþ1Þ�1p ; ð14Þ
where sp is the Golden p-Proportion, p2{0,1,2,3, . . .}.It is clear that this general principle includes in itself the ‘‘Dichotomy Principle’’ (3) and the classical ‘‘Golden Sec-
tion Principle’’ (6) as the special cases for p = 0 and p = 1.
3.3. The Generalized Principle of the Golden Section in modern philosophy
Recently the Byelorussian philosopher Eduardo Soroko developed very original approach to structural harmony of
systems [5]. He used the generalized Golden Sections and formulated his famous ‘‘Law of Structural Harmony of Sys-
tems’’ as follows:
The Generalized Golden Sections are invariants, which allow natural systems in process of their self-organization tofind harmonious structure, stationary regime of their existence, structural and functional stability.
Table 1
p 1 2 3 4 5 6 7
sp 1.618 1.465 1.380 1.324 1.285 1.255 1.232
bp 0.6180 0.6823 0.7245 0.7549 0.7781 0.7965 0.8117
If we sum the binomial coefficients of the ‘‘deformed’’ Pascal triangle we will come unexpectedly to the Fibonacci
numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . !If we shift every row of the initial Pascal triangle by p columns (p = 0,1,2,3, . . .) to the right with respect to the pre-
ceding column and then sum the binomial coefficients of the new ‘‘deformed’’ Pascal triangle by columns we will come
to the numerical sequence that is expressed by the following recurrence relation:
F pðnÞ ¼ F pðn� 1Þ þ F pðn� p � 1Þ for n > p þ 1; ð16Þ
F pð1Þ ¼ F pð2Þ ¼ ¼ F pðp þ 1Þ ¼ 1: ð17Þ
Notice that the recurrence relation (16) for the initial terms (17) gives an infinite number of new numerical sequences.
Moreover, the ‘‘binary’’ sequence 1, 2, 4, 8, 16, . . . , is the special case of this sequence for p = 0 and the classical Fibo-
nacci numbers 1, 1, 2, 3, 5, 8, 13, . . . are the special case of this sequence for p = 1! We will name the numerical sequence
generated by (16), (17) by Fibonacci p-numbers.
Let us consider now the ratio of the two adjacent Fibonacci p-numbers Fp(n)/Fp(n � 1) for the case n ! 1. It is
proved [9] that this ratio strives to the Golden p-Proportion sp!
It is impossible to overestimate methodological importance of the deep mathematical connection of the Generalized
Principle of the Golden Section given by (14) with Pascal triangle and binomial coefficients. It is clear that this connec-
tion can become the beginning for reappraisal of many branches of modern mathematics and theoretical physics where
combinatorial relations play important role, in particular, probability theory and statistical laws.
5. A new measurement theory based on the Generalized Principle of the Golden Section
5.1. The first optimization problem in measurement theory
As is well known measurement theory has a long history. Its origin is connected to the ‘‘incommensurable segments’’
discovery made by Pythagoreans at investigation of the ratio of the square diagonal to its side. This discovery caused
the first crisis in mathematics foundations and resulted to appearance of irrational numbers.
In 1202 the first optimization problem appeared in measurement theory. The famous Italian mathematician Fibo-
nacci was the author of this problem. This problem is called the ‘‘problem of choosing the best system of standard
weights’’ or Bashet–Mendellev�s problem (in the Russian mathematical literature [9]).
The essence of the problem consists in the following [9]. Let it be necessary to weigh any unknown weight Q in the
range from 0 up to Qmax using n standard weights
fq1; q2; . . . ; qng; ð18Þ
where q1 = 1 is a measurement unit; qi = ki · qi; ki is any natural number.
It is clear that the maximum weight Qmax is equal to the sum of all standard weights, i.e.
Qmax ¼ q1 þ q2 þ þ qn ¼ ðk1 þ k2 þ þ knÞq1: ð19Þ
Then it appears the problem to find the optimal system of the standard weights, i.e. such standard weight system (18),
which ensures the maximum value of Qmax given by (18) among all possible variants of (18). In this case we have to
choose such variant of the standard weights (18) that it would be possible to compose any multiple by q1 weight Q using
the standard weights (18) taking each of them separately.
As is well known there are two variants of this problem [9]. For the former case we can place the standard weights
only on the free cup of the balance, for the latter case we can place them on two cups of the balance.
The optimal solution for the former case is given by the ‘‘binary’’ system of standard weights, i.e.
f1; 2; 4; 8; 16; . . . ; 2n�1g: ð20Þ
Notice that the measurement algorithm based on the ‘‘binary’’ system ‘‘generates’’ so-called ‘‘binary’’ measurement
algorithm that is used widely in the measurement practice. Notice that the ‘‘binary’’ algorithm ‘‘generates’’ the ‘‘binary’’
number system that underlies modern computers and information technology.
5.2. The ‘‘Asymmetry Principle of Measurement’’
The analysis of the ‘‘binary’’ algorithm by using the balance model (Fig. 2) allows to discover one surprising mea-
surement property having a general character for all thinkable measuring that are reduced to the comparison of the
measurable weight Q with any standard weights.
Q 2n-1
2n-1<Q 202n-1 2n-2
...
...
add
Q
2n-1
2n-1 Q 202n-1 2n-2
...
...
remove
2n-2202n-1 2n-2
...
...
add
(a)
(b)
(c)
Fig. 2. The ‘‘Asymmetry Principle of Measurement’’.
6. The ‘‘Asymmetry Principle’’ of the living nature
6.1. ‘‘Asymmetry’’ of ‘‘rabbits reproduction’’
As is well known the classical Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . were introduced by the famous Italian
mathematician Leonardo Pisano Fibonacci in the 13 century at the solution of the well-known ‘‘rabbits reproduction’’
problem. This problem and its solution (Fibonacci numbers) have two important corollaries for development of modern
science. First of all the ‘‘rabbits reproduction’’ problem gave an origin of the mathematical theory of biological popula-
tions [34]. Second, the Fibonacci recurrence relation gave an origin of the recurrence relation method that is one of the
most important methods of combinatorial analysis [1–3].
Let us remind ourselves that the ‘‘Law of rabbit reproduction’’ is reduced to the following rule. Each ‘‘mature rabbit
couple’’ A births a ‘‘newborn rabbit couple’’ B during one month. The ‘‘newborn rabbit couple’’ matures during one
month and one more in one month starts to bring one rabbit couple. Thus, ‘‘maturing’’ the newborn rabbits, that is,
their transformation into mature couple is realized in 1 month. We can model a process of ‘‘rabbits reproduction’’ by
using two passages:
A ! AB ð22Þ
B ! A ð23Þ
Notice that the passage (22) models a process of the ‘‘birth’’ of the ‘‘newborn rabbit couple’’ B and the passage (23)
models a process of ‘‘maturing’’ the ‘‘newborn rabbit couple’’ B. The passage (22) reflects ‘‘asymmetry’’ of ‘‘rabbits
reproduction’’ because the ‘‘mature rabbit couple’’ A turns into the two non-identical couples, the ‘‘mature rabbit cou-
ple’’ A and the ‘‘newborn rabbit couple’’ B.
6.2. The generalized ‘‘Asymmetry Principle’’ of the living nature
Using the model of ‘‘rabbits reproduction’’ one may generalize the ‘‘Asymmetry Principle’’ of the Living Nature. To
this end we will take the non-negative number p P 0 and consider the following problem:
‘‘Let us suppose that in the enclosed place there is a couple of rabbits (female and male) in the first day of January. This
rabbit couple reproduces a new rabbit�s couple in the first day of February and then in the first day of each next month. Each
newborn rabbit�s couple becomes mature in p months and then gives a life to the new rabbit�s couple each month after. There
is a question: how much rabbit�s couples will be in the enclosed place in one year, that is, in 12 months from the beginning of
reproduction?’’
It is clear that for the case p = 1 the generalized variant of the ‘‘rabbit reproduction’’ problem coincides with the
classical ‘‘rabbits reproduction’’ problem [1–3].
Note that the case p = 0 corresponds to that idealized situation when ‘‘rabbits’’ become ‘‘mature’’ right away after
birth. One may model this case using the passage:
A ! AA ð24Þ
It is clear that that the passage (24) reflects ‘‘symmetry’’ of ‘‘rabbits reproduction’’ when the ‘‘mature rabbit couple’’ A
turns into the two identical ‘‘mature rabbit couple’’ A. It is easy to show that for this case the ‘‘rabbits’’ are reproduced
according to the ‘‘Dichotomy principle’’, that is, the ‘‘rabbits’’ double each month: 1, 2, 4, 8, 16, 32, . . .Let us consider now the case p > 0. Let us analyze more in detail the formulated above problem taking into consid-
eration the new conditions of ‘‘rabbit reproduction’’. It is clear that ‘‘reproduction process’’ is described by more com-
plex system of ‘‘passages’’ describing the ‘‘reproduction process’’. Really, let A and B be the couples of ‘‘mature’’ and
‘‘newborn’’ rabbits respectively. Then the passage (22) models a process of monthly appearance of the ‘‘newborn cou-
ple’’ B from each ‘‘mature couple’’ A.
Let us consider now a process of transformation of the ‘‘newborn couple’’ B into the ‘‘mature couple’’ A. It is evident
that during ‘‘maturing’’ the ‘‘newborn couple’’ B passes through intermediate stages corresponding to each month:
For example, for the case p = 2 a process of transformation of the ‘‘newborn couple’’ into the ‘‘mature couple’’ is
described by the following system of ‘‘passages’’:
B ! B1 ð26Þ
B1 ! A ð27Þ
Then, taking into consideration (22), (26), (27) the process of ‘‘rabbit reproduction’’ for the case p = 2 can be rep-
resented with help of Table 2.
Notice that the column A gives a number of the ‘‘mature couples’’ for each stage of reproduction, the column B gives
a number of the ‘‘newborn couples’’, the column B1 gives a number of the ‘‘newborn couples’’ being in the state B1 the
column A + B + B1 gives a general number of the rabbits couples for each stage of reproduction.
The analysis of the numerical sequences in each column
A : 1; 1; 1; 2; 3; 4; 6; 9; 13; 19; . . .
B : 0; 1; 1; 1; 2; 3; 4; 6; 9; 13; . . .
B1 : 0; 0; 1; 1; 1; 2; 3; 4; 6; 9; 13; . . .
Aþ Bþ B1 : 1; 2; 3; 4; 6; 9; 13; 19; . . .
shows that they are subordinated to one and the same regularity: each number of the sequence is equal to the sum of the
preceding number and the number distant from the latter in two positions. But we know the Fibonacci 2-numbers given
by the recurrence relation (16) are subordinated to that regularity!
If we carry out similar considerations for the general case of p we will come to conclusion that the Fibonacci p-num-
bers given by the recurrence relation (16) would be a solution of the generalized variant of the ‘‘rabbit reproduction’’
problem! And they reflect the ‘‘Generalized Asymmetry Principle’’ of the Living Nature.
At first sight, the above formulation of generalized problem of ‘‘rabbit reproduction’’ has no real ‘‘physical’’ sense.
But we will not hurry with conclusions! The paper [34] is devoted to application of the generalized Fibonacci numbers
for modeling of biological cell growth. In the paper it is affirmed that ‘‘in kinetic analysis of cell growth, the assumption is
usually made that sell division yields two daughter cells symmetrically. The essence of the semi-conservative replication of
chromosomal DNA implies complete identity between daughter cells. Nonetheless, in bacteria, insects, nematodes, and
plants, cell division is regularly asymmetric, with spatial and functional differences between the two products of division . . .Mechanism of asymmetric division include cytoplasmic and membrane localization of specific proteins or of messenger
RNA, differential methylation of the two strands of DNA in a chromosome, asymmetric segregation of centrioles and mito-
chondria, and bipolar differences in the spindle apparatus in mitosis’’. In [34] it is analyzed the models of cell growth based
on the Fibonacci 2- and 3-numbers.
In the summery the authors of [34] made the following conclusion: ‘‘Binary cell division is regularly asymmetric in
most species. Growth by asymmetric binary division may be represented by the generalized Fibonacci equation . . . Our mod-
els, for the first time at the single cell level, provide rational bases for the occurrence of Fibonacci and other recursive phyl-
lotaxis and patterning in biology, founded on the occurrence of regular asymmetry of binary division’’.
Thus, the results of the paper [34] show that the world of biology is based on the ‘‘Generalized Principle of the
Let us consider the infinite set of the ‘‘standard segments’’ based on the Golden p-Ratio sp:
Gp ¼ fsnpg; ð31Þ
where n = 0, ±1, ±2, ±3, . . . ; snp are the Golden p-Ratios powers connected among themselves by the identity (11).
The set (31) ‘‘generates’’ the following constructive method of the real number A representation called the code of
the Golden p-Proportion:
A ¼Xi
aisip; ð32Þ
where ai 2 {0,1} and i = 0, ±1, ±2, ±3, . . .Notice that the positional number systems (32) were introduced by the present paper author in 1980 u. in the paper [35]
and called the Codes of the Golden p-Proportion. A theory of these number systems was developed in author�s book [11].
Let us consider the partial cases of the number representation (32). It is clear that for the case p = 0 the formula (32)
is reduced to (7). Finally, let us consider the case p ! 1. For this case it is possible to show, that sp ! 1; it means that
the positional representation (32) is reduced to the Euclidean definition (29).
Notice that for the case p > 0 the radix sp of the positional number system (32) is irrational number. It means that we
have came to the number systems with irrational radices that a principally new class of the positional number systems.
Notice that for the case p = 1 the number system (32) is reduced to the Bergman�s number system (8) that had been
introduced by the American mathematician George Bergman in 1957 [4]. Notice that Bergman�s number system (8)
is the first number system with irrational radix in the history of mathematics.
It follows from the given consideration that the positional representation (32) is very wide generalization of the clas-
sical binary number system (7), Bergman�s number system (8) and the Euclidean definition (29) that are partial cases of
the general representation (32).
Possibly the number system with irrational radix (8) developed by George Bergman in 1957 and its generalization
given by (32) are the most important mathematical discoveries in the field of number systems after discovery of posi-
tional principle of number representation (Babylon, 2000 B.C.) and decimal number system (India, 5th century).
7.5. Some properties of the number systems with irrational radices
Note that the expression (32) divides the set of real numbers into two non-overlapping subsets, the ‘‘constructive’’
real numbers that can be represented in the form of the final sum (32) and all the rest real numbers that cannot be rep-
resented in the form of the sum (32) and are called the ‘‘non-constructive’’ real numbers. Such approach to real numbers
is distinguished radically from the classical approach when the set of real numbers is divided into rational and irrational
numbers.
Really, all powers of the Golden p-Proportion of the kind sip (i = 0,±1,±2,±3, . . .) that are irrational numbers can be
represented in the form (32), that is, they refer to the subset of the ‘‘constructive’’ numbers. For example,
s1p ¼ 10 s�1p ¼ 0:1
s2p ¼ 100 s�2p ¼ 0:01
s3p ¼ 1000 s�3p ¼ 0:001:
It follows from the definition (32) that all real numbers that are the sums of the Golden p-Proportion powers are
‘‘constructive’’ numbers of the kind (32). For example, according to (32) the real number A ¼ s2p þ s�1p þ s�3
p can be rep-
resented as the following binary code combination:
A ¼ 100; 101:
Notice that a possibility of representation of some irrational numbers (the powers of the Golden p-Proportion and
their sums) in the form of the final totality of bits is the first unusual property of the number systems (32).
7.6. Representation of natural numbers
Let us consider the representation of natural numbers in the form (32):
N ¼Xi
aisip ð33Þ
where N is some natural number, sp is the radix of number system (33), ai2{0,1},i = 0, ±1, ±2, ±3, . . .
This result has a great importance for mathematics and general science. As all natural numbers can be represented in
the form (33) it means that one may formulate new scientific doctrine ‘‘Everything is the Golden p-Proportion’’ instead
the Pythagorian doctrine ‘‘Everything is a number’’.
7.7. Z-property of natural numbers
Let us prove that the new definition of real number (32), (33) based on the Generalized Principle of the Golden Sec-
tion can become a source of new number-theoretical results. The Z-property [14] of natural numbers is one of them.
For the proof of this property let us represent some natural number N in Bergman�s number system:
N ¼Xi
aisi ð34Þ
where s ¼ 1þffiffi5
p
2¼ 1:618 and i = 0, ±1, ±2, ±3, . . .
The expression (34) is called the s-code of natural number N.
It is well known the following formula in the Fibonacci numbers theory [1–3]:
si ¼ Li þ F i
ffiffiffi5
p
2: ð35Þ
where Fi and Li are Fibonacci and Lucas numbers, s is the Golden Proportion and i = 0, ±1, ±2, ±3, . . . It is consideredthat the formula (35) had been deduced in the 19 century by the French mathematician Binet.
Using formula (35) it is easy to prove the following theorem [14].
Theorem 1 (Z-property of natural numbers). If we represent some natural number N in Bergman�s number system (34)
and then replace every power of the Golden Ratio si in the expression (34) by the Fibonacci number Fi, where the index i
takes its values from the set {0,±1,±2,±3, . . .}, then the sum arising as result of such replacing is equal to 0 identically
independently on the initial natural number N, that is,
Xi
aiF i ¼ 0:
In [14] it is proved the following theorem that is a generalization of Theorem 1.
Theorem 2 (Zp-property of natural numbers). If we represent some natural number N in the Golden p-Ratio code
N ¼P
iaisip, where p > 0 and i = 0, ±1, ±2, ±3, . . . , and replace each power of the Golden p-Ratio in it with the
corresponding Fibonacci p-number Fp(i) then the sum arising at this replacement is identically equal to 0 independently on
the initial natural number N, that is
Xi
aiF pðiÞ ¼ 0:
Note that the properties given by Theorems 1 and 2 are valid only for natural numbers! It means that our investi-
gations have led us to discovery of the new property of natural numbers called Z- or Zp-property (from the word
‘‘Zero’’). And this unusual property could be discovered in mathematics only after discovery of number systems with
irrational radices given by (8) and (32).
8. Fibonacci and ‘‘Golden’’ matrices
8.1. Q-matrix
In the last decades the theory of the Fibonacci numbers was supplemented by the theory of so-called Q-matrix [3].
The latter presents by itself the 2 · 2 matrix of the following form:
Q ¼1 1
1 0
� �: ð36Þ
It is easy to calculate the Q-matrix determinant, which equals to �1.
Let us represent the matrix (37) in the form of the two matrices given for the even (n = 2k) and odd (n = 2k + 1)
values of the index n:
Q2k ¼F 2kþ1 F 2k
F 2k F 2k�1
� �; ð49Þ
Q2kþ1 ¼F 2kþ2 F 2kþ1
F 2kþ1 F 2k
� �: ð50Þ
Using the correlation (46) it is possible to represent the matrices (49), (50) in the terms of the symmetric hyperbolic
Fibonacci functions (44), (45):
Q2k ¼cFsð2k þ 1Þ sFsð2kÞ
sFsð2kÞ cFsð2k � 1Þ
!; ð51Þ
Q2kþ1 ¼sFsð2k þ 2Þ cFsð2k þ 1ÞcFsð2k þ 1Þ sFsð2kÞ
� �; ð52Þ
where k is the discrete variable, k = 0,±1,±2,±3, . . .And now we will replace the discrete variable k in the matrices (51), (52) by the continues variable x:
Q2x ¼cFsð2xþ 1Þ sFsð2xÞsFsð2xÞ cFsð2x� 1Þ
� �; ð53Þ
Q2xþ1 ¼sFsð2xþ 2Þ cFsð2xþ 1ÞcFsð2xþ 1Þ sFsð2xÞ
� �: ð54Þ
It is clear that the matrices (53), (54) are a generalization of the Q-matrix (37) for continues domain. They have a few
unusual mathematical properties. For example, for the case x ¼ 14the matrix (53) takes the following form:
Q12 ¼
ffiffiffiffiQ
p¼
cFsð32Þ sFsð1
2Þ
sFsð12Þ cFsð� 1
2Þ
!: ð55Þ
It is impossible to imagine that means the ‘‘root square from the Q-matrix’’ but this ‘‘Fibonacci�s fantasy’’ follows fromthe expression (55).
And if we calculate the determinants of the matrices (53), (54) then using (47), (48) we will come to one more ‘‘fan-
tastic’’ results that are valid for any value of the continues variable x:
DetQ2x ¼ 1; ð56Þ
DetQ2xþ1 ¼ �1: ð57Þ
Thus, Fibonacci numbers, Fibonacci p-numbers, and the hyperbolic Fibonacci functions have led us to very unusual
matrices given by (37), (41), (53), (54). Their extraordinary nature consists in the fact that according to (42), (56), (57)
their determinants are equal 1 or �1 independently on the continues variable x!
9. The generalized Principle of the Golden Section in computer engineering
9.1. Fibonacci codes and arithmetic
Fibonacci�s algorithms considered above are isomorphic to the following positional representation of natural num-
A positional representation of the natural number N in the form (58) is called Fibonacci p-code. The abridged nota-
tion of the Fibonacci p-code (58) has the following form:
N ¼ an an�1 ai a1: ð59Þ
Notice that the notion of the Fibonacci p-code includes an infinite amount of different methods of the binary rep-
resentations as every number p ‘‘generates’’ its own Fibonacci p-code (p = 0,1,2,3, . . .).Let us consider the partial cases of the Fibonacci p-code (58). For the case p = 0 the Fibonacci p-numbers given by
(14), (15) are reduced to the binary numbers: 1, 2, 4, 8, 16, 32, 64, 128, . . . , 2i�1, . . . , that is,
F 0ðiÞ ¼ 2i�1: ð60Þ
Substituting (60) into the formula (58) we have:
N ¼ an2n�1 þ an�12
n�2 þ þ ai2i�1 þ þ a12
0: ð61Þ
It means that for the case p = 0 the Fibonacci p-code (58) is reduced to the classical binary representation of natural
numbers.
Let p = 1. For this case the Fibonacci p-numbers are reduced to the classical Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13,
21, 34, . . . , Fn. It is clear that the Fibonacci p-code (58) for this case is reduced to so-called Zeckendorf�s representation[2]:
N ¼ anF n þ an�1F n�1 þ þ aiF i þ þ a1F 1 ð62Þ
Let us consider now the partial case p = 1. In this case every Fibonacci p-number is equal to 1 identically, that is, for
any integer i we have
F pðiÞ ¼ 1:
Then the sum (58) takes the following form:
N ¼ 1þ 1þ þ 1|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}N
: ð63Þ
Thus, the Fibonacci p-code given by (58) is a very wide generalization of the binary code (61) and Zeckendorf�s rep-resentation (62) that are the partial cases of the Fibonacci p-code (58) for the cases p = 0 and p = 1 respectively. On the
other hand, the Fibonacci p-code (58) includes in itself so-called the ‘‘unitary’’ code (63) as another extreme case for
p = 1.
9.2. Surprising analogies between Fibonacci and genetic codes
Among the biological concepts, which are well formalized and have a level of the general scientific importance, the
genetic code takes a special place. Discovering the well-known fact of striking simplicity of basic principles of the ge-
netic code falls into the major modern discoveries of human science. This simplicity consists in the fact that the inher-
itable information is encoded by the texts from the three-alphabetic words—triplets or codonums compounded on the
basis of the alphabet that consists of the four characters being the nitrogen bases: A (adenine), C (cytosine), G (guan-
ine), T (thiamine). The given recording system is essentially unified for all boundless set of miscellaneous alive organ-
isms and is called genetic code.
It is found that by using three-alphabetic triplets or codonums we can encode the 21 items including the 20 amino
acids and one additional item called stop-codonum (sign of the punctuation) encoded by triplets. Then there are 43 = 64
different combinations from four on three nitrogen bases used for the coding the 21 items. In this connection some of
the 21 items are encoded at once by several triplets. It is called as a degeneracy of the genetic code. Finding of conformity
between triplets and amino acids (or signs of the punctuation) is customary treated as decryption of genetic code.
Let us consider now the 6-digit Fibonacci 1-code (Zeckendorf�s representation) that uses six Fibonacci numbers 1, 1,
10. The generalized principle of the Golden Section in electrical engineering
10.1. The ‘‘binary’’ resistor divider
In electrical engineering practice so-called resistor dividers intended for division of electric currents and voltages in
the given ratio are widely used. One variant of such divider is shown in Fig. 3.
The resistor divider in Fig. 3 consists of the ‘‘horizontal’’ resistors of the kind R1 and R3 and the ‘‘vertical’’ resistors
of the kind R2. The resistors of the divider are connected between themselves by the ‘‘connecting points’’ 0, 1, 2, 3, 4.
Each point connects three resistors that form together the resistor section. Note that Fig. 3 demonstrates the resistor
divider that consists of the 5 resistor sections. In general, a number of resistor sections can be equal to n (n = 1,2,3, . . .).First of all, we note that the parallel connection of the resistors R2 and R3 to the right of the ‘‘connecting point’’ 0
and to the left of the ‘‘connecting point’’ 4 can be replaced by the equivalent resistor with the resistance, which can be
calculated according to the law on the resistor parallel connection:
Re1 ¼R2� R3R2þ R3
: ð70Þ
Taking into consideration (70) it is easy to find the equivalent resistance of the resistor section to the right of the
‘‘connecting point’’ 1 and to the left of the ‘‘connecting point’’ 3:
Re2 ¼ R1 þ Re1: ð71Þ
In dependence on the choice of the resistance values of the resistors R1, R2, R3 we can realize the different coeffi-
cients of the current or voltage division. Let us consider the so-called ‘‘binary’’ divider that consists of the following
resistors: R1 = R; R2 = R3 = 2R, where R is some standard resistance value. For this case the expressions (70), (71) take
the following values:
Re1 ¼ R; Re2 ¼ 2R: ð72Þ
Then, taking into consideration (70)–(72) we can find that the equivalent resistance of the resistor circuit to the left
or to the right of any ‘‘connecting point’’ 0, 1, 2, 3, 4 is equal to 2R. It means that the equivalent resistance of the divider
in any of the ‘‘connecting points’’ 0, 1, 2, 3, 4 can be calculated as the resistance of the parallel connection of the three
resistors of the value 2R. Using the electrical circuit laws we can calculate the equivalent resistance of the divider in each
Let us connect now the generator of electric current I to one of the ‘‘connecting points’’, for example, to the point 2.
Then according to Ohm�s law the following electric voltage will appear in this point:
U ¼ 2
3RI : ð74Þ
Let us calculate the electrical voltages in the ‘‘connecting points’’ 3 and 1 that are adjacent to the point 2. It is easy to
show that the voltage transmission coefficient between the adjacent ‘‘connecting points’’ is equal to 12. It means that the
‘‘binary’’ divider fits very well to the binary number system and this fact is a reason of wide use of the ‘‘binary’’ resistor
divider in modern digit-to-analog and analog-to-digit converters and modern measurement systems.
10.2. The ‘‘golden’’ resistor dividers
Let us take the values of the resistors in Fig. 3 as the following:
R1 ¼ s�pp R; R2 ¼ spþ1
p R; R3 ¼ spR; ð75Þ
where sp is the Golden p-Ratio, p 2 {0,1,2,3, . . .}.It is clear that the divider in Fig. 3 gives an infinite number of the different resistor dividers as each value of p ‘‘gen-
erates’’ a new divider. In particular, for the case p = 0 we have: s0 = 2 and the divider is reduced to the classical ‘‘binary’’
divider.
For the case p = 1 the resistors R1, R2, R3 take the following values:
R1 ¼ s�1R; R2 ¼ s2R; R3 ¼ sR; ð76Þ
where s ¼ 1þffiffi5
p
2is the Golden Ratio.
And now we will investigate the basic electrical properties of the ‘‘golden’’ resistor divider (Fig. 3) given by (76). To
this end we will use the following properties of the Golden p-Ratio:
sp ¼ 1þ s�pp ; ð77Þ
spþ2p ¼ spþ1
p þ sp: ð78Þ
Let us calculate the equivalent resistance of the resistive circuit of the divider to the left and to the right from the
‘‘connecting points’’ 0 and 4 using the expression (78):
Re1 ¼R2� R3R2þ R3
¼spþ1p R� spR
spþ1p Rþ spR
¼ R: ð79Þ
Note that we simplified the expression (79) using the mathematical identity (78).
Using (71) and (77) it is possible to calculate the equivalent resistance of Re2:
Re2 ¼ s�pp Rþ R ¼ spR: ð80Þ
Thus, according to (80) the equivalent resistance of the resistor circuit of the divider to the left or to the right of any
of the ‘‘connecting points’’ 0, 1, 2, 3, 4 is equal to spR where sp is the Golden p-Ratio. This fact can be used for cal-
culation of the equivalent resistance Re3 of the divider in the ’’connecting points‘‘ 0, 1, 2, 3, 4. In fact, the equivalent
resistance Re3 can be calculated as the resistance of the electrical circuit that consists of the parallel connection of the
‘‘vertical’’ resistor R2 ¼ spþ1p R and the two ‘‘lateral’’ resistors with the resistance spR. But as the equivalent resistance of
the parallel connection of the resistors R2 ¼ spþ1p R and R3 = spR is equal to R according to (79) then the equivalent
resistance Re3 of the divider in any of the ‘‘connecting points’’ can be calculated by the formula:
Re3 ¼spR� RspRþ R
¼ spsp þ 1
R ¼ 1
1þ s�1p
R: ð81Þ
Note that for the case p = 0 (the ‘‘binary’’ divider) we have: sp = s0 = 2 and then the expression (81) is reduced to
(74).
Let us calculate the voltage transmission coefficient between the adjacent ‘‘connecting points’’ of the ‘‘golden’’ divi-
der. To this end we will connect the generator of the electric current I to one of the ‘‘connecting points’’, for example, to
the point 2. Then according to Ohm�s law the following electrical voltage appears in this point
Let us calculate the electrical voltage in the adjacent ‘‘connecting points’’ 3 b 1. The voltages in the points 3 and 1 can
be calculated as a result of linking the voltage U given by (82) to the resistor circuit that consists of the sequential con-
nection of the ‘‘horizontal’’ resistor RI ¼ s�pp R and the resistor circuit with the equivalent resistance R. Then, for this
case the electrical current that appears in the resistor circuit to the left and to the right of the ‘‘connecting point’’ 2 will
be equal to
UR1þ R
¼ Uðs�p
p þ 1ÞR ¼ UspR
: ð83Þ
If we multiply the electrical current given by (83) by the equivalent resistance R we will obtain the following value of
the electrical voltage in the adjacent ‘‘connecting points’’ 3 and 1:
Usp: ð84Þ
It means that the voltage transmission coefficient between the adjacent ‘‘connecting points’’ of the ‘‘golden’’ divider in
Fig. 3 is equal to the reciprocal of the Golden p-Ratio!
Thus, the ‘‘golden’’ resistor dividers based on the Golden p-Ratios sp are quite real electrical circuits. It is clear that
the stated above theory of the ‘‘golden’’ resistor dividers could become a new source for development of so-called ‘‘dig-
ital metrology’’ and analog-to-digit and digit-to-analog converters. It is important to emphasize that new class of resis-
tor dividers are based on the Generalized Principle of the Golden Section!
10.3. The ‘‘golden’’ digit-to-analog and analog-to-digit converters
The electrical circuit of the «golden» DAC based on the ‘‘golden’’ resistor divider in Fig. 3 is shown in Fig. 4. Note
that the ‘‘golden’’ DAC in Fig. 4 consists of the five resistor sections. However the number of the DAC resistor sections
may be increased to some arbitrary n by extending the resistor divider to the left and to the right.
The ‘‘golden’’ DAC contains the five (n in the general case) generators of the standard electrical current I0 and the
five (n in the general case) electrical current keys K0 � K4. The key states are controlled by the binary digits of the Gold-
en p-Ratio code a4a3a2a1a0. For the case ai = 1 the key Ki is closed, for the case ai = 0 it is open (i = 0,1,2, . . .n).One can show that the closed key Ki results in the following voltage at the ith point of the resistor divider:
As the potential Ui is passed from the ith point to the (i + 1)th point with the transmission coefficient 1spthe following
voltage appears at the DAC output:
Uout ¼bpI0R
sn�l�1p
¼bpI0R
sn�1p
� sip:
Using the superposition principle it is easy to show that the Golden p-Ratio code an�1an�2 . . .a0 results in the fol-
lowing voltage Uout:
U ¼ Bp
Xn�1
i¼0
alsip; ð85Þ
where
Bp ¼bpI0R
sn�1p
:
It follows from (85) that the electrical circuit in the Fig. 4 converts the Golden p-Ratio code (31) into the electrical
voltage Uout with regard to the constant coefficient Bp.
The ‘‘golden’’ DAC in Fig. 4 is a basis of so-called self-correcting ADC�s [7,16] that are ‘‘insensitive’’ to the DAC
‘‘technological’’ and ‘‘temperature’’ errors.
Note that the considered above ‘‘golden’’ ADC and DAC are the basis of new projects in the field of ‘‘digital metro-
logy’’ and measurement systems.
11. The generalized principle of the Golden Section in communication systems
11.1. A new approach to Shannon’s theory of communication
Let us demonstrate application of the Generalized Principle of the Golden Section to Shannon�s communication the-
ory. Let us consider the traditional ‘‘channel without noise’’ (Fig. 5).
As is well known, a channel capacity is the most important characteristic of the channel in Fig. 5. According to the
communication theory developed by the American scientist Claude Shannon the channel capacity is given by the fol-
lowing expression:
C ¼ supR ¼ limT!1
1
TIðn; gÞ; ð86Þ
where R is a speed of information transmission in the channel, T is a time of information transmission, I(n,g) is averageinformation quantity transmitted through the channel during the time T, n and g are input and output messages
accordingly.
Let us consider so-called ‘‘symmetric’’ channel. In it the information is transmitted by bits 0 and 1 having equal
length DT. To calculate the channel capacity under the formula (86), it is necessary to know the transmission time T
and to calculate the information quantity I(n,g). If we know a number n of bits of the given message we can calculate
the time of T under the following trivial formula:
T ¼ nDT : ð87Þ
To calculate I(n,g) we have to remind that for the ‘‘channel without noise’’ we have:
Iðn; gÞ ¼ log2NðT Þ; ð88Þ
where N(T) is a number of all possible discrete messages that could be transmitted through the channel during the time
T. Then taking into consideration (88) we can write the expression (86) as follows:
3. Measurement theory. As is well known, measurement theory based on Eudoxus–Archimedus� and Cantor�s axi-oms is one of the fundamental theories of mathematics. From this point of view algorithmic measurement theory
[9,10,12] has fundamental interest for mathematics development. The Italian mathematician Leonardo Pisano
(Fibonacci) became famous for two mathematical discoveries, viz. the problem of choosing the best system of
the standard weights (Bachet–Mendeleev�s problem), when he invented the binary number system, and the prob-
lem of rabbits reproduction, which gave the Fibonacci numbers. The ‘‘Asymmetry Principle of Measurement’’,
which combined both Fibonacci�s problems, showed the deep inner connection between the ‘‘rabbits reproduc-
tion’’ and Bashet–Mendeleev�s problems. This analogy has fundamental interest for applications of algorithmic
measurement theory, in particular for the mathematical theory of biological populations.
4. Theory of elementary functions. The hyperbolic Fibonacci and Lucas functions [38–40] are a new class of elemen-
tary functions that have ‘‘strategic’’ importance for the development of modern mathematics and physics. One
may assume the origin of the following cosmologic theories from this approach: (1) Lobatchevski–Fibonacci
geometry as the Fibonacci interpretation of Lobatchevski geometry; (2) Minkovski–Fibonacci geometry as the
Fibonacci interpretation of Einstein�s theory of relativity.
5. Fibonacci numbers theory. Harmony mathematics creates new stimulus for development of the Fibonacci number
theory [1–3]. First of all new recurrence relations of the algorithmic measurement theory ‘‘generate’’ new numer-
ical sequences that expand the range of Fibonacci�s research. On the other hand, hyperbolic Fibonacci and Lucas
functions [38–40] transform the Fibonacci numbers theory into ‘‘continued’’ theory that allows to apply to the
Fibonacci numbers theory mathematical apparatus of the ‘‘continued’’ mathematics, in particular differentiation
and integration.
6. Matrix theory. The Fibonacci matrices (41) and the ‘‘golden’’ matrices of the kind (53), (54) have unique math-
ematical properties (42), (56), (57). Studying these matrices is an interesting direction in matrix theory. A search
of applications of these matrices in theoretical physics is one of the important directions of the physics research.
7. Physics. The articles [19–32] exhibit a substantial interest of modern theoretical physics and the physics of high
energy particles in the Golden Section. The works of Shechtman, Butusov, Mauldin and William, El Naschie, and
Vladimirov show that it is impossible to imagine the future progress in physical and cosmological research with-
out the Golden Section. The famous Russian physicist-theoretician Prof. Vladimirov (Moscow University) fin-
ished his book ‘‘Metaphysics’’ [29] with the following words: ‘‘Thus, it is possible to assert that in the theory of
electroweak interactions there are relations that are approximately coincident with the ‘‘Golden Section’’ that play
an important role in the various areas of science and art’’.
8. Philosophy. ‘‘The Law of Structural Harmony of Systems’’ by Eduard Soroko [5] is possibly one of the most out-
standing modern philosophical achievements. In essence Soroko�s Law is a brilliant confirmation of the General-
ized Golden Section Principle application to self-organizing systems.
9. Botanic. The new geometric theory of the botanic phyllotaxis phenomenon developed in the book [6] by the
Ukrainian architect Oleg Bodnar (Bodnar�s theory) is a brilliant confirmation of the hyperbolic Fibonacci and
Lucas functions application for simulation of phyllotaxis phenomenon.
10. Biology. Cell growth is one of the actual biological problems. It was proven in [34] that binary cell division is
regularly asymmetric and based on the Fibonacci p-numbers that follows from the Pascal triangle. It means that
the binary cell division is based on the Generalized Principle of the Golden Section.
11. Medicine. The Russian biologist Viktor Tsvetkov asserts in his book ‘‘Heart, the Golden Section and Symmetry’’
[48] that mammal�s cardiogram is subordinated to the Golden Section Principle. In Tsvetkov�s opinion, the orga-nization of the ‘‘golden’’ cardiac cycle is a result of a long evolution of mammals in the direction of optimization
of their structure and functions.
12. Computers. Fibonacci and ‘‘golden’’ computer arithmetic that was developed in [9,11,41–45] can become the basis
of new computer projects (Fibonacci noise-tolerant processor [43], ‘‘Golden’’ fault-tolerant ternary mirror-sym-
metrical processor [45] and so on).
13. Measurement systems. The ‘‘golden’’ resistor dividers are the electrical basis of the ‘‘golden’’ analog-to-digit and
digit-to-analog converters [43] and could lead to new measurement system projects.
14. Communication systems. Fibonacci matrices (41) and the ‘‘golden’’ matrices (53), (54) are the mathematical basis
for new a coding theory [47] that could be used effectively for error detection and correction in communication
channels and for cryptographic protection of communication systems.
15. Museum of Harmony and the Golden Section [49,50] is a unique History, Science, Art and Nature museum that
has no analogy in world culture. The Museum is a collection of all Science, Art and Natural works based on the
Golden Section.
16. Reform of mathematical education. Traditionally so-called ‘‘Elementary Mathematics’’ that was developed in
ancient time underlies the basis of mathematical education in secondary schools. Johannes Kepler once said: